The Rayleigh model describes scattering of light by particles whose size is much smaller than the wavelength of the light. This scatter model is applicable to many situations, the most common of which is scattering of sun light in the Earth’s atmosphere.

In the Rayleigh model, the distribution of scattered light for an input beam which is unpolarized is given by:

p(q,l) = 0.375*(1 + cos²q)/l⁴

where l is the wavelength and q is the angle of the scattered ray with respect to the specular ray; q = 0 degrees refers to scattering along the specular ray in the forward direction, and q = 180 refers to scattering along the specular ray in the backward direction.

(The coefficient of 0.375 is needed to ensure that the Total Integrated Scatter (TIS = ∫ p(q,l) sinq dq, with the limits of integration from 0 to p) is equal to unity.)

The Rayleigh scattering distribution may be applied to any non-sequential volume object using a user defined DLL (RAYLEIGH.DLL) that has been provided with the ZEMAX installation (\ZEMAX\Objects\DLL\BulkScatter\). Tests have been conducted to investigate both the distribution of power and the angular distribution of the scattering model in ZEMAX, and they confirm that the Rayleigh model has been properly implemented. These tests are identical to those described in the Knowledge Base article “Using the Henyey-Greenstein Distribution to Model Bulk Scattering”.

A test has also been conducted to validate the wavelength dependence of the scattering model. The test file (Rayleigh_WaveTest.ZMX) is provided in the .ZIP file located at the end of this article.

The test design consists of a source launching rays at normal incidence to a rectangular volume in which the Rayleigh scattering model has been applied. Inputs to the DLL are the reference wavelength and the transmission:



The reference wavelength (l₀) is specified in microns; it is the wavelength associated with the input value of scattering mean free path (M), as provided in the “Mean Path” input of the dialog box. Thus, in the example shown above the scattering mean free path is 1.0 mm (M is always specified in lens units) at a wavelength of 0.55 mm. The mean free path varies with wavelength as:

M(l) = M(l₀)*(l/l₀)⁴

The transmission parameter describes how much of the input power is attenuated during scattering.

Rays that pass through the rectangular volume are then recorded on a Detector Rectangle object. For arbitrary values of the mean free path (M) and the length of the volume (L), some rays may pass through the volume without undergoing bulk scattering. The fraction of “unscattered” rays can be determined from the fact that rays which travel a distance x within the volume have an integrated probability of having been scattered given by:

p(x) = 1.0 - e^-x/M

as described in the chapter of the ZEMAX manual entitled “Non-Sequential Components”. Thus, the integrated probability of a ray traveling a distance x within the volume and not undergoing scattering is given by 1 - p(x) = e^-x/M. Setting x = L, we find that the probability of a ray traveling through the full volume and not being scattered is e^-L/M.

In our example, L = 1.0 mm and M = 1.0 mm at a reference wavelength of 0.55 mm, which is also the wavelength of the source rays. For this case, the fraction of unscattered rays is simply e^-1.0/1.0 = e^-1 = 0.368.

A ray trace may be performed, in which the results are saved to a ray database file:



This file is used to filter the ray data that is obtained from the Detector Viewer. Specifically, we are only interested in those rays which do not undergo bulk scattering. The filter string “!B2” can be used to isolate such rays:



The total number of rays which do not undergo scattering and hit the detector is about 36800 (the actual number will vary for any given ray trace):



(When the filter is applied, all of the associated rays only strike the center pixel of the detector, which is why emission is not seen elsewhere). According to the equations given in the previous section, when the length of the volume equals the input mean free path and the source wavelength is the same as the reference wavelength – as they are in this case – the fraction of unscattered rays is 0.368. In this test 100,000 rays were launched towards the volume, so we would expect 0.368*100,000 = 36800 rays to pass through the volume without undergoing scattering. This is indeed the case!

Since the mean-free path for scattering varies with wavelength, the fraction of unscattered rays will also vary with wavelength. For example, if the input mean free path is 1.0 mm at a reference wavelength of 0.55 mm, then at a wavelength of 0.65 mm the actual mean free path is 1.95 mm. If the length of the volume is still 1.0 mm, then the fraction of unscattered rays is e^-1.0/1.95 = 0.599. For 100,000 source rays, the number of rays which pass through the volume without scattering should be 59900.

If we change the system wavelength (= source ray wavelength) to 0.65 mm, and re-run the ray trace (making sure to save the results to a .ZRD file, so that a filter string may be applied in the Detector Viewer), we find that the number of rays which pass through the volume unscattered is about 60000:



The actual number will vary from ray trace to ray trace, but in all cases the agreement with the expected value is excellent (within < 1%). This has been confirmed for multiple values of l and L, indicating that the wavelength variation of the Rayleigh model is correctly implemented in the DLL.

The Rayleigh model describes the distribution of light scattered from particles which are much smaller than the wavelength of the light. A user-defined DLL which allows users to apply this bulk scattering model to any non-sequential volume is included with the ZEMAX installation. The known angular distribution and wavelength variation of the Rayleigh model have been incorporated in the DLL, and have been tested for accuracy.